L(s) = 1 | + (−1.05 + 1.83i)2-s + (−1.24 − 2.14i)4-s + (−1.34 − 2.32i)5-s + (−0.486 + 0.842i)7-s + 1.01·8-s + 5.67·10-s + (−0.158 + 0.274i)11-s + (0.757 + 1.31i)13-s + (−1.02 − 1.78i)14-s + (1.40 − 2.43i)16-s − 1.17·17-s + 6.22·19-s + (−3.32 + 5.76i)20-s + (−0.335 − 0.580i)22-s + (1.08 + 1.87i)23-s + ⋯ |
L(s) = 1 | + (−0.748 + 1.29i)2-s + (−0.620 − 1.07i)4-s + (−0.599 − 1.03i)5-s + (−0.183 + 0.318i)7-s + 0.359·8-s + 1.79·10-s + (−0.0477 + 0.0827i)11-s + (0.209 + 0.363i)13-s + (−0.275 − 0.476i)14-s + (0.351 − 0.608i)16-s − 0.284·17-s + 1.42·19-s + (−0.743 + 1.28i)20-s + (−0.0715 − 0.123i)22-s + (0.225 + 0.390i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.370233 + 0.641263i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.370233 + 0.641263i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (1.05 - 1.83i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.34 + 2.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.486 - 0.842i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.158 - 0.274i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.757 - 1.31i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 + (-1.08 - 1.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.20 - 3.81i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.33 - 7.51i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 + (-2.92 - 5.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.79 - 4.84i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.23 - 2.14i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (0.862 + 1.49i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.507 - 0.878i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.428 + 0.741i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + (-5.60 + 9.71i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.34 + 4.05i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + (-2.77 + 4.80i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34825548123885169750956915492, −9.263140620024646259237145315507, −8.926671395904681518159716824223, −8.071048187867451435502491601135, −7.35927634411976562476862134363, −6.47196631654845252732713230648, −5.42241843063361283140282662324, −4.71136522508940279676033546619, −3.23993956856859068121671646211, −1.09588014930606698960080929459,
0.61391397650232916310165313531, 2.31218274410756333973911624608, 3.26057461790872879897873707219, 3.98389269901978673705985206526, 5.64213702165844004233337554964, 6.86393047771686290129406160942, 7.66897493515549359922717554809, 8.575928517986612756512254199570, 9.569668327277850238158680513230, 10.26362393606000322266413296303